Get ready for a big one. I think Barry should have another profession. The idea of a cop superhero is good and Dick Grayson could be one for example. But Barry being a cop was just a random profession they chose to fill in the blanks. He couldn't be just another scientist so they went with something a little different but still connected to science. Clark Kent as a reporter and Superman makes total sense. Matt Murdoch as a lawyer and Daredevil also makes total sense. Barry Allen and company as a Police Scientist is just random. The best Flash stories you could write would in no way be helped by him being a cop. It would just become a part of the plot out of contrivance.
To be clear, I haven't found a good solution to this. I definitely think he needs a secret Identity, and I want something humble. Maybe he could be an electrician if you want him to be an everyman. I kind of like Danica Williams working at a museum but I suppose Hawkman already took that and from a lore perspective it makes much more sense for him. Being a banker kind of fits Barry's whole personality but I'm not sure how it would work, specially with him disappearing all the time. Heck maybe a programmer could work. He could easily dress up and do something when nobody was looking and come back an hour later. The idea of the internet being a big deal in his life isn't necessary but it could also work with him running around the city and all. Or maybe he could be an accountant.
If you started everything from scratch on the basis that Barry Allen, with the same personality, is the Flash, would you gain anything by making him a CSI? Everything he does as a CSI is what he would do as the Flash. I mean Batman also investigates a crime scene, takes evidence and analyses it in a lab. Barry would also do the aame things if he was just the Flash. From a storytelling perspective you are just repeating the same job.
Other Super hero cops are characters that become superheroes because the law won't let them go far enough, or because they are worried that criminals will come after their family if they know who they are (but this is something also common in superheroes that aren't cops). If Barry revealed his secret identity and was hired by the police department he would do everything he already does as the Flash, because a cop with super powers will always be able to do things that a normal cop can't.
Flash is the type of hero where him being the flash really overwhelms his stories. Very few stories in the last 40 years had his detective work as a central point of the plot (with Wally it wasn't that much different even though he wasn't a CSI). When you read a Flash story it usually either has to do with a crisis or villain coming after him personally or attacking the city in clear view, or it's about a fantastical threat that no cop would ever be involved in.
So does it make sense to give him such an active job since it doesn't really get used? And even if it got used, what would be the difference between Mirror Master robbing a bank and Flash hearing about it on the police radio and using his super speed to investigate the crime scene as the Flash, versus him being a cop assigned to the case. You could just remove that part and everything would be the same. He would even still have a relationship with the cops simply for being the Flash.
Compare that to Clark Kent, Matt Murdoch, sometimes Bruce Wayne and his company, Black Lightning as a school principal and Peter Parker as a school teacher. Their jobs either put them in situations they wouldn't be in as superheroes, and that somehow help them in their cases or add another level of personal dilemas or perspective to whatever crisis they are solving.